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Cross-validation Bandwidth Matrices for Multivariate Kernel Density Estimation

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  • TARN DUONG
  • MARTIN L. HAZELTON
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    Abstract

    The performance of multivariate kernel density estimates depends crucially on the choice of bandwidth matrix, but progress towards developing good bandwidth matrix selectors has been relatively slow. In particular, previous studies of cross-validation (CV) methods have been restricted to biased and unbiased CV selection of diagonal bandwidth matrices. However, for certain types of target density the use of full (i.e. unconstrained) bandwidth matrices offers the potential for significantly improved density estimation. In this paper, we generalize earlier work from diagonal to full bandwidth matrices, and develop a smooth cross-validation (SCV) methodology for multivariate data. We consider optimization of the SCV technique with respect to a pilot bandwidth matrix. All the CV methods are studied using asymptotic analysis, simulation experiments and real data analysis. The results suggest that SCV for full bandwidth matrices is the most reliable of the CV methods. We also observe that experience from the univariate setting can sometimes be a misleading guide for understanding bandwidth selection in the multivariate case. Copyright 2005 Board of the Foundation of the Scandinavian Journal of Statistics..

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    File URL: http://www.blackwell-synergy.com/doi/abs/10.1111/j.1467-9469.2005.00445.x
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    Bibliographic Info

    Article provided by Danish Society for Theoretical Statistics & Finnish Statistical Society & Norwegian Statistical Association & Swedish Statistical Association in its journal Scandinavian Journal of Statistics.

    Volume (Year): 32 (2005)
    Issue (Month): 3 ()
    Pages: 485-506

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    Handle: RePEc:bla:scjsta:v:32:y:2005:i:3:p:485-506

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    Cited by:
    1. Seppo Pulkkinen & Marko Mäkelä & Napsu Karmitsa, 2013. "A continuation approach to mode-finding of multivariate Gaussian mixtures and kernel density estimates," Journal of Global Optimization, Springer, vol. 56(2), pages 459-487, June.
    2. J. Chacón & T. Duong, 2010. "Multivariate plug-in bandwidth selection with unconstrained pilot bandwidth matrices," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer, vol. 19(2), pages 375-398, August.
    3. Rob J. Hyndman & Han Lin Shang, 2008. "Rainbow plots, Bagplots and Boxplots for Functional Data," Monash Econometrics and Business Statistics Working Papers 9/08, Monash University, Department of Econometrics and Business Statistics.
    4. Billings, Stephen B. & Johnson, Erik B., 2012. "A non-parametric test for industrial specialization," Journal of Urban Economics, Elsevier, vol. 71(3), pages 312-331.
    5. Shuowen Hu & D.S. Poskitt & Xibin Zhang, 2010. "Bayesian Adaptive Bandwidth Kernel Density Estimation of Irregular Multivariate Distributions," Monash Econometrics and Business Statistics Working Papers 21/10, Monash University, Department of Econometrics and Business Statistics.
    6. Duong, Tarn & Cowling, Arianna & Koch, Inge & Wand, M.P., 2008. "Feature significance for multivariate kernel density estimation," Computational Statistics & Data Analysis, Elsevier, vol. 52(9), pages 4225-4242, May.
    7. Nicolai, R.P. & Koning, A.J., 2006. "A general framework for statistical inference on discrete event systems," Econometric Institute Research Papers EI 2006-45, Erasmus University Rotterdam, Erasmus School of Economics (ESE), Econometric Institute.
    8. Schoch, Tobias & Staub, Kaspar & Pfister, Christian, 2012. "Social inequality and the biological standard of living: An anthropometric analysis of Swiss conscription data, 1875–1950," Economics & Human Biology, Elsevier, vol. 10(2), pages 154-173.
    9. Horová, Ivana & Koláček, Jan & Vopatová, Kamila, 2013. "Full bandwidth matrix selectors for gradient kernel density estimate," Computational Statistics & Data Analysis, Elsevier, vol. 57(1), pages 364-376.
    10. Filippone, Maurizio & Sanguinetti, Guido, 2011. "Approximate inference of the bandwidth in multivariate kernel density estimation," Computational Statistics & Data Analysis, Elsevier, vol. 55(12), pages 3104-3122, December.

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